Abstract
Collectively coupling molecular ensembles to a cavity has been demonstrated to modify chemical reactions akin to catalysis. Theoretically understanding this experimental finding remains an important challenge. In particular the role of quantum effects in such setups is an open question of fundamental and practical interest. Theoretical descriptions often neglect quantum entanglement between nuclear and electrophotonic degrees of freedom, e.g., by computing Ehrenfest dynamics. Here we discover that disorder can strongly enhance the buildup of this entanglement on short timescales after incoherent photoexcitation. We find that this can have direct consequences for nuclear coordinate dynamics. We analyze this phenomenon in a disordered HolsteinTavisCummings model, a minimal toy model that includes all fundamental degrees of freedom. Using a numerical technique based on matrix product states we simulate the exact quantum dynamics of more than 100 molecules. Our results highlight the importance of beyond BornOppenheimer theories in polaritonic chemistry.
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Introduction
Polaritonic chemistry, or the modification of chemical reactivity using effects of cavity quantum electrodynamics (cavityQED), is an emerging field of research at the interface of quantum chemistry and physics^{1,2,3,4,5,6,7,8,9}. Experiments have demonstrated that a collective coupling of electronic^{10,11,12,13,14,15,16,17,18} or vibrational^{19,20,21,22,23} transitions of large ensembles of molecules to confined nonlocal electromagnetic fields can provide means to control chemical reactivity. Many experiments have achieved a collective strong coupling regime, where the cavity and the molecules can coherently exchange energy at a rate faster than their decay processes. In such scenarios, the cavitymolecule system has to be considered as one entity with new polaritonic eigenstates, which are collective superpositions of photonic and molecular degrees of freedom. Identifying the underlying mechanisms of collective cavitymodified chemistry remains to be a major challenge. A theoretical understanding of the problem requires to solve complex quantum manybody dynamics in large systems with coupled electronic, photonic, and vibrational degrees of freedom.
Numerically computing the collective time evolution of all degrees of freedom in polaritonic chemistry is an important—yet extremely challenging—task for understanding chemical reaction dynamics, which has been attempted at different levels of approximations. For small systems, the Schrödinger equation can be solved directly^{24} or using quantum chemistry tools such as multiconfigurational timedependent HartreeFock methods^{25}. Density functional theory can be used for ab initio simulations of a few realistic molecules^{26}. For larger systems, stronger approximations are needed. Standard approaches are based on the BornOppenheimer approximation. In the BornOppenheimer approximation, electrophotonic dynamics are treated as instantaneous compared to nuclear dynamics so that polaritonic (and dark) potential energy surfaces can be computed^{27}. On these adiabatic potential energy surfaces, nuclear dynamics can then be computed. However, this method neglects nonadiabatic couplings between potential energy surfaces and thus fails if the separation between potential energy surfaces becomes small, as is often the case in polaritonic chemistry^{5,28,29}. In order to include the nonadiabatic couplings, two common methods are fewest switches surface hopping^{30,31,32} or meanfield Ehrenfest dynamics^{33,34}. Ehrenfest dynamics assumes a product state between nuclear and electrophotonic degrees of freedom, completely neglecting any entanglement between them. As a consequence, such entanglement can serve as a measure for the validity of approximations relying on the separability of nuclear and electrophotonic degrees of freedom, and more generally the complexity of the dynamics. Alternatively, vibrationally dressed electronic molecular states have been computed using polaron ansätze^{35,36,37,38,39}. Such approximations allow for the computation of effective transfer rates between these states for very large molecule numbers, but, by construction, they cannot compute the entangled vibrationalelectrophotonic dynamics.
The role of quantum effects in molecular dynamics is also a fundamentally interesting research question^{40,41,42,43}. Entanglement is often used to determine the importance of quantum effects by quantifying quantum correlations without classical equivalent^{44,45,46}. In this context, the entanglement between electronic and nuclear degrees of freedom of molecules has been previously studied for single molecules^{47,48}. For cavitycoupled molecules, it is known that a collective cavitycoupling can strongly suppress this entanglement by reducing vibronic couplings, an effect termed “polaron decoupling”^{35,36}. However, this effect neglects local disorder in the electronic level spacings of individual molecules, which is generally present in organic polaritonic setups^{49,50,51}.
Recently, matrix product states (more broadly: tensor networks) have been suggested to numerically tackle dynamics in polaritonic chemistry^{52} also for larger system sizes. A matrix product state (MPS) can be thought of as a generalization of a product state, which by definition does not include any entanglement, into a larger space with small but finite entanglement. The entanglement of an MPS is limited by a socalled bond dimension, which can be systematically increased until convergence is reached^{53}. Since excessively large entanglement rarely plays an important role in physical dynamics, MPS simulations often become numerically exact. By construction, MPS concepts provide direct access for studying the entanglement dynamics of a system, and they have been used in that context extensively, e.g., for spinchain or Hubbardtype models in manybody physics^{45,46}.
Here, using this numerical approach we study the femtosecondscale dynamics of more than 100 molecules with electronic transitions collectively strongly coupled to a cavity mode (electronic strong coupling) after an incoherent photoexcitation (see Fig. 1 for a sketch). We analyze a minimal disordered version of the HolsteinTavisCummings (HTC) model^{35,54,55}, which despite its simplicity includes the main ingredients for microscopically understanding modifications of vibrational dynamics in cavitycoupled molecular ensembles, in particular in organic systems. We find that disorder enhances excitation transfer from the initially excited state to a number of molecules selected by a resonance condition (see Fig. 1b, c)^{56,57,58}. This leads to coherent outofphase oscillations of the vibrational modes of these molecules. As a consequence, disorder enhances entanglement between vibrations and electronic degrees of freedom severalfold (see the sketch in Fig. 1d). This effect is largest in a regime where disorder is energetically comparable to collective cavitycouplings. Importantly, we find that the disorderinduced focused excitation transfer to a few molecules leads to an enhanced cavitymodified vibrational dynamics on the singlemolecule level, compared to a disorderless scenario where the excitation is diluted among all coupled molecules equally. This effect crucially depends on whether the initial incoherent excitation is absorbed by a single molecule or the cavity, and we analyze both scenarios (see Fig. 1b). We further relate large entanglement to modifications of the shape of the nuclear wave packets, which become broadened and nonGaussian. In this respect, the vibrational entanglement may have direct consequences for chemical processes.
Results and discussion
Theoretical model
We consider a system of N toymodel molecules coupled to a singlemode optical cavity, i.e., a disordered version of the HTC model^{35,54,55}. Here, each molecule has two electronic energy levels. Different nuclear equilibrium configurations in the ground and excitedstate result in two displaced harmonic onedimensional potential energy surfaces as shown in Fig. 1a. We further include an inhomogeneous broadening, i.e., a disorder of the electronic energy stemming from random energy spacings of the electronic levels^{59}, typically induced by the environment in experiments. The disordered HTC Hamiltonian reads^{35}
The coupling of the cavity is described by the TavisCummings (TC) Hamiltonian, which, in a frame rotating at the cavity frequency ω_{C}, reads (ℏ = 1 throughout this paper)
where \(\hat{a}\) is the destruction operator for a cavity photon, \({\hat{\sigma }}_{n}^{\pm }\) are the raising/lowering operators for the electronic level of the nth molecule. Δ = ω − ω_{C} is the detuning between the electronic transition frequency at the Condon point ω and ω_{C}, chosen to be Δ = 0 in the remainder of this paper. The coupling strength of a single molecule to the cavity is given by \(g\equiv {g}_{{{{{{{{\rm{c}}}}}}}}}/\sqrt{N}\). In the singleexcitation Hilbert space considered here, the TC Hamiltonian has two polariton eigenstates \(\left\pm \right\rangle ={\hat{a}}^{{{{\dagger}}} }/\sqrt{2}\pm {\sum }_{n}{\hat{\sigma }}_{n}^{+}/\sqrt{2N}{\left0\right\rangle }_{{{{{{{{\rm{exc+ph}}}}}}}}}\) for the ground state \({\left0\right\rangle }_{{{{{{{{\rm{exc+ph}}}}}}}}}\) without any excitations, split by the Rabi splitting of 2g_{c}. The other N − 1 eigenstates are degenerate dark states with zero energy.
The nuclear coordinates are described by harmonic potentials
where \({\hat{b}}_{n}\) is the lowering operator of the nth molecule and ν the molecular oscillation frequency. The eigenstates of \({\hat{H}}_{{{{{{{{\rm{vib}}}}}}}}}\) are Fock states \({\prod }_{n}{({\hat{b}}_{n}^{{{{\dagger}}} })}^{{a}_{n}}\left{0}_{{{{{{{{\rm{vib}}}}}}}}}\right\rangle\) with a_{n} vibrational quanta on the nth molecule and the (undisplaced) total vibrational ground state \(\left{0}_{{{{{{{{\rm{vib}}}}}}}}}\right\rangle\). We define dimensionless oscillator position and momentum variables as \({\hat{x}}_{n}=({\hat{b}}_{n}+{\hat{b}}_{n}^{{{{\dagger}}} })/\sqrt{2}\) and \({\hat{p}}_{n}={{{{{{{\rm{i}}}}}}}}({\hat{b}}_{n}{\hat{b}}_{n}^{{{{\dagger}}} })/\sqrt{2}\), respectively.
The nuclear coordinate of each molecule is coupled to its electronic state by a Holstein coupling
This corresponds to a shift of the excited state potential energy surface. The dimensionless HuangRhys factor λ^{2} quantifies the minimum of the excited state harmonic potential at position \(\sqrt{2}\lambda\) with energy ω − λ^{2}ν.
Finally, we include disorder by
where ϵ_{n} = ω_{n} − ω is the deviation of the electronic transition energy of the nth molecule from the mean. We take the ϵ_{n} as independent, normally distributed random variables with mean 0 and variance W^{2}. Here, we focus on energy disorder in the molecular level spacing, which is a fundamental characteristic of typical organic setups^{49,50,51}. In addition, in real setups typically also the cavitycoupling g is disordered, e.g., due to fluctuations in the molecular orientations and the cavity mode profile. Here, we find that this additional disorder only leads to minor modifications for the entanglement dynamics (see Supplementary Note 1), and we thus do not further consider it in the following.
Dynamics and entanglement
In the following, we analyze the shorttime Hamiltonian dynamics on the scale of a single nuclear vibration period, 0 ≤ t ≤ 2π/ν for two different initial states. In one case, a single molecule (n = 1) is excited by the incoherent absorption of a photon, i.e., we consider the initial state \(\left{\psi }_{0}^{{{{{{{{\rm{m}}}}}}}}}\right\rangle ={\hat{\sigma }}_{1}^{+}{\left0\right\rangle }_{{{{{{{{\rm{ph}}}}}}}}}{\left0\right\rangle }_{{{{{{{{\rm{exc}}}}}}}}}{\left0\right\rangle }_{{{{{{{{\rm{vib}}}}}}}}}\) (Fig. 1b, left). In the other case the photon is incoherently absorbed by the cavity, \(\left{\psi }_{0}^{{{{{{{{\rm{c}}}}}}}}}\right\rangle ={\hat{a}}^{{{{\dagger}}} }{\left0\right\rangle }_{{{{{{{{\rm{ph}}}}}}}}}{\left0\right\rangle }_{{{{{{{{\rm{exc}}}}}}}}}{\left0\right\rangle }_{{{{{{{{\rm{vib}}}}}}}}}\) (Fig. 1b, right). Here, \({\left0\right\rangle }_{{{{{{{{\rm{exc,vib,ph}}}}}}}}}\) denote the respective ground states of the bare electronic, vibrational, and photonic Hamiltonian.
In order to analyze entanglement between electrophotonic and nuclear degrees of freedom, we separate the full Hilbert space as \({{{{{{{\mathcal{H}}}}}}}}={{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{ph}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{exc}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{vib}}}}}}}}}\) into three subHilbert spaces for the cavity photon, electronic excitations, and vibrations, respectively. For a pure state \(\left\psi \right\rangle\), the entanglement between the two subsystems \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{ph}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{exc}}}}}}}}}\) and \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{vib}}}}}}}}}\) can be quantified by the von Neumann entropy of either subsystem,^{45,46} e.g.,
where \({\hat{\rho }}_{{{{{{{{\rm{vib}}}}}}}}}\) is the reduced density matrix which can be obtained from the state \(\left\psi \right\rangle\) by tracing over \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{ph}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{exc}}}}}}}}}\): \({\hat{\rho }}_{{{{{{{{\rm{vib}}}}}}}}}={{{{{{{{\rm{Tr}}}}}}}}}_{{{{{{{{\rm{ph}}}}}}}}+{{{{{{{\rm{exc}}}}}}}}}(\psi \rangle \langle \psi )\). In the case of a product state (or meanfield) assumption, the state of the system would be assumed to factorize throughout the evolution of the system
In this scenario, \({\hat{\rho }}_{{{{{{{{\rm{vib}}}}}}}}}(t)={\phi }_{{{{{{{{\rm{vib}}}}}}}}}(t)\rangle \langle {\phi }_{{{{{{{{\rm{vib}}}}}}}}}(t)\) and S_{vib}(t) = 0 at all times. An entangled state \(\psi \rangle\) is a linear superposition of many such terms, resulting in S_{vib} > 0. The von Neumann entropy S_{vib} can be readily computed in the MPS framework (see Methods). It is noteworthy that this product state assumption is equivalent to the one made in meanfield Ehrenfest dynamics, where in addition the nuclear motion is treated classically. Since in our case the nuclear wavefunction always stays in a coherent state and thus follows classical equations of motion, our product state results are equivalent to meanfield Ehrenfest results.
Parameter regimes
We choose parameter values that are motivated by a setup with Rhodamine 800, for which strong coupling has been demonstrated^{60}, and which has been previously considered in tensor network studies of strong coupling experiments^{52}. In particular, we set 0.1 ≤ λ ≤ 0.5 and ν = 0.3g_{c}. For an experimentally demonstrated vacuum Rabi splitting of 2g_{c} = 700 meV^{10}, this corresponds to ν = 105 meV and reorganization energies 1 meV ≲ λ^{2}ν ≲ 26 meV, similar to measured values^{61}. Thermal excitation fractions \(\sim \exp (\nu /{k}_{{{{{{{{\rm{B}}}}}}}}}T)\) are negligible at room temperature (k_{B}T ≈ 26 meV). In particular, in the case of Rhodamine 800 the level spacing is ~2 eV^{60}, for which the Rabi splitting of 700 meV (g_{c}/ω ~ 18%) falls just into a regime beyond the onset of ultrastrong coupling effects. Note that we still do not include counterrotating terms in order to derive general results for strong coupling, valid for other molecules with larger level spacings ω or smaller Rabisplittings g_{c}. Smaller Rabisplittings will generally lead to an increased ratio of λν/g_{c} and thus additionally induced mixing with vibrational degrees of freedom over the Holstein coupling term \({\hat{H}}_{{{{{{{{\rm{H}}}}}}}}}\).
For our case of λν ≪ ν ≪ g_{c}, the Hamiltonian Eq. (1) can be categorized into strong (W ≪ g_{c}) and weak (W ≫ g_{c}) coupling regimes depending on the relative magnitude of \({\hat{H}}_{{{{{{{{\rm{TC}}}}}}}}}\) and \({\hat{H}}_{{{{{{{{\rm{dis}}}}}}}}}\). The strong coupling regime features polaritonic and dark eigenstates of \({\hat{H}}_{{{{{{{{\rm{TC}}}}}}}}}\) which are mixed perturbatively (Fig. 1b). In perturbation theory, we find that “gray” states \(\leftd\right\rangle\) acquire photocontributions of \({\sum }_{d}{\langle d{1}_{{{{{{{{\rm{ph}}}}}}}}}\rangle }^{2} \sim {\lambda }^{2}{\nu }^{2}/(2{g}_{{{{{{{{\rm{c}}}}}}}}}^{2})\) and \({\sum }_{d}{\langle d{1}_{{{{{{{{\rm{ph}}}}}}}}}\rangle }^{2}\approx {W}^{2}/{g}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) due to small vibronic coupling and disorder, respectively^{49,50,58,59} (see Supplementary Notes 2 and 3). In the weak coupling regime, polariton states cease to exist and all eigenstates are structurally similar to the “gray” states in Fig. 1a. We vary 0 ≤ W ≤ 1.5g_{c} analyzing both weak and strong coupling scenarios. The timescale of vibrational evolution t ~ 2π/ν corresponds to tens of femtoseconds, and can be faster than dissipative mechanisms which we do not include explicitly. For quality factors Q ≳ 1000 which have e.g., been achieved for distributed Bragg reflectors, cavity decay is negligible on these timescales^{62}. Similarly, relaxation of molecular excitation into vibrational or electromagnetic reservoirs typically occurs on even slower timescales of picoseconds or nanoseconds, respectively^{55}. In fact, on a microscopic level the coherent dynamics due to disorder and vibronic coupling terms \({\hat{H}}_{{{{{{{{\rm{dis}}}}}}}}}\) and \({\hat{H}}_{{{{{{{{\rm{H}}}}}}}}}\) that we simulate here can be considered as one of the mechanisms responsible for electronic dephasing.
Summary of results
Figure 2 visualizes the main feature of the entanglement and vibrational dynamics after initial molecular (\(\left{\psi }_{0}^{{{{{{{{\rm{m}}}}}}}}}\right\rangle\), panels a, c, e), and cavity (\(\left{\psi }_{0}^{{{{{{{{\rm{c}}}}}}}}}\right\rangle\), panels b, d, f) excitation. Strikingly, in both scenarios, we find that increasing the disorder in the range 0 ≤ W ≤ g_{c} leads to a drastically enhanced entanglement entropy buildup, seen in the evolution of the disorderaveraged entropy \(\overline{{S}_{{{{{{{{\rm{vib}}}}}}}}}}\) in Fig. 2a, b. For W = 0, the entanglement entropy remains below values of one, and, in the cavity excitation case only, exhibits oscillatory features which we can attribute to collective Rabi oscillation due to a predominant excitation transfer to the polariton states. For W > g_{c}/2 those features disappear and we observe a strong increase to a maximum value at t ~ 2π/ν and t ~ π/ν in Fig. 2a, b, respectively. This entanglement buildup is matched by modifications of the phase space dynamics (Fig. 2c, d) and the shape of the probability distribution P_{i}(x_{i}, t) (Fig. 2e, f) of the nuclear coordinate. Below, we will relate all three effects to disorderenhanced excitation transfer. We will see that neither the entanglement buildup nor the distribution shape changes can be captured by a product state assumption Eq. (7), i.e., they go beyond the meanfield Ehrenfest dynamics.
Fig. 2c, d shows the phase space dynamics of the disorderaveraged expectation values \(\overline{{x}_{i}}\) and \(\overline{{p}_{i}}\) of the nuclear coordinate position and momentum operators \({\hat{x}}_{i}\) and \({\hat{p}}_{i}\) on molecule i, respectively. In the molecular excitation case in Fig. 2c, we observe phase space circles for the initially excited molecule (i = 1). In the disorderless case W = 0, we find an oscillation around the displaced equilibrium position of the excited state oscillator, \(\sqrt{2}\lambda \approx 0.57\). In this case, the evolution is very close to the nocavity scenario (gray dashed line), for which we obtain a perfect circle around \(\sqrt{2}\lambda\) corresponding to the usual coherent harmonic oscillator evolution.
However, the situation changes drastically for W > 0. Now, the centers of the phase space circles dynamically shift to smaller values of \(\overline{{x}_{1}}\). We note that this behavior can be rationalized without requiring the large entanglement buildup seen in Fig. 2a. Assuming the product state ansatz from Eq. (7), one would expect that the Holstein term \({\hat{H}}_{{{{{{{{\rm{H}}}}}}}}}\) [Eq. (4)] leads to an effective excited state oscillator equilibrium position of \(\sqrt{2}\lambda \langle {\hat{\sigma }}_{1}^{+}{\hat{\sigma }}_{1}^{}\rangle\) and thus effectively to a timedependent shift of the minimum depending on \(\langle {\sigma }_{1}^{+}{\hat{\sigma }}_{1}^{}\rangle (t)\). For W = 0 the initial state \(\left{\psi }_{0}^{{{{{{{{\rm{m}}}}}}}}}\right\rangle\) is almost a dark eigenstate of \({\hat{H}}_{{{{{{{{\rm{TC}}}}}}}}}\), such that cavity induced excitation transfer is strongly suppressed and the excitation remains on the molecule, \(\langle {\hat{\sigma }}_{1}^{+}{\hat{\sigma }}_{1}^{}\rangle (t) \sim 1\). In contrast, for finite disorder W > 0, the excitation transfer is significantly enhanced and we perturbatively derive \(1\langle {\hat{\sigma }}_{1}^{+}{\hat{\sigma }}_{1}^{}\rangle (t) \sim Wt/N\) for g ≪ W ≪ g_{c} (see Supplementary Note 4). This is in qualitative agreement with recent results predicting that disorder can enhance excitation transfer in models without vibrations^{56,57,58}.
For an initial cavity excitation \(\left{\psi }_{0}^{{{{{{{{\rm{c}}}}}}}}}\right\rangle\), the phase space evolution traces much smaller circles. As expected, for W = 0 the phase space evolution is approximately centered at \(\overline{{x}_{i}} \sim \sqrt{2}\lambda /(2N)\), and exhibits oscillations at polariton Rabi frequencies. For increasing disorder W → g_{c}, the center of the circle now shifts in the opposite direction compared to Fig. 2c, to roughly twice the value \(\overline{{x}_{i}}\to \sqrt{2}\lambda /N\). This can again be rationalized by looking at the evolution of the expected local molecule excitations \(\langle {\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{}\rangle (t)\). For W = λν = 0, the hybrid nature of the polariton states induces Rabi oscillations between the initial cavity photon state and a collective excitation of all molecules, such that for each molecule the excitation fraction oscillates according to \(\langle {\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{}\rangle (t)={\cos }^{2}({g}_{{{{{{{{\rm{c}}}}}}}}}t)/N\), leading to the observed phase space evolution for W = 0. Finite disorder, however, leads to a photocontribution of all dark states (perturbatively \(\sim\!\! {W}^{2}/{g}_{{{{{{{{\rm{c}}}}}}}}}^{2}\)). Therefore, excitations are now irreversibly (on our timescale of interest) transferred from the cavity to individual molecules, and one thus expects a disorderaveraged excitation population on each molecule \(\overline{\langle {\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{}\rangle }\to 1/N\) for sufficiently large W. This explains the observed shift. Importantly, the vibrational dynamics only depend on the transfer of excitations from cavity to molecules, independently of whether the molecular excitations are part of polaritons or dark states, such that the relevant timescale for excitation transfer remains 1/g_{c}.
Figure 2e, f shows the probability distribution P_{i}(x_{i}, t) of the nuclear coordinate at time t = 2π/ν. Without cavity, at this time the distribution is a Gaussian centered at x_{1} = 0 with variance 1/2, corresponding to a coherent state (gray dashed line). We find that in a cavity and for W = 0, the distribution is extremely close to the nocavity scenario. For increasing W, however, for the molecular excitation \(\left{\psi }_{0}^{{{{{{{{\rm{m}}}}}}}}}\right\rangle\), the distribution of the nuclear coordinate of the initially excited molecule P_{1}(x_{1}, t = 2π/ν) clearly shifts to smaller values of x_{1}. In addition, the distribution broadens and acquires an asymmetric shape in Fig. 2e. For an initial cavity excitation \(\left{\psi }_{0}^{{{{{{{{\rm{c}}}}}}}}}\right\rangle\), we observe that finite W leads to modifications in the tails of the distribution only, i.e., for large values of x_{i} (Fig. 2f). Note that the tail modifications in Fig. 2f seem very small, since a single molecule only receives a ~1/N contribution of the excitation energy (here, N = 100). However, below we see that the cumulative effect on the wavefunction shape can still have important consequences for many molecules.
Excitation transfer dynamics
In Fig. 3, we now exemplify the connection between the timedependent local molecular excitation and the phase space evolution for an intermediate disorder strength W = g_{c}/2 microscopically. Figure 3a shows the vibrational evolution of each of the 100 molecules for a single disorder realization, after exciting one molecule initially (line with star: excited molecule, other red/cyan/yellow lines: 99 initially unexcited molecules). Strikingly, we observe that the molecules whose dynamics are modified most strongly correspond to the ones with random energy very close to the initially excited one. The reason for this is seen in comparison with Fig. 3b where we plot the excitation numbers of the molecules at t = 2π/ν as a function of their random energy offset ϵ_{i}. There we identify the molecules with the strongest phase space modification (cyan square and orange diamond), and the initially excited ones (blue star and vertical line). Crucially, the excitation fraction of these molecules, and thus their phase space dynamics, is much larger than for the disorderless case with W = 0 (gray lines in Fig. 3a, b, barely visible in a). The same behavior is generally seen also after disorderaveraging (see inset). We attribute a visible asymmetry towards smaller energies in Fig. 3b to additional resonances with states of higher vibrational energies. It is also interesting to point out that in contrast to the initially excited molecule, the phase space variables of the other molecules generally do not complete one revolution until t = 2π/ν (Fig. 3a), and all molecular oscillators evolve outofphase.
A similar picture presents itself when initially exciting the cavity mode (Fig. 3c, d). The excitation is again primarily transferred from the cavity to several molecules, but now with energies ϵ_{i} ~ ± g_{c} close to resonance with the bare polaritons in the strong coupling regime. These molecules acquire much larger excitation fractions than the no disorder reference (gray line in Fig. 3d). As a result, molecular oscillations of molecules with an energy offset ϵ_{n} ~ ± g_{c} are most strongly modified, as confirmed in Fig. 3c.
We can deduce the following microscopic picture from our analysis in Fig. 3: While in the W = 0 case the initial excitation is generally diluted throughout the system, disorder W > 0 leads to a strongly enhanced excitation transfer to a few molecules in the energetic vicinity of either the initially excited molecule or the polariton states, depending on the scenario (as sketched in Fig. 1b). In a product state picture, this then modifies the vibrational dynamics of those molecules depending on the amount of local excitation, \(\langle {\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{}\rangle\). However, the product state assumption contradicts the buildup of large vibrational entanglement seen in Fig. 2a, b. Rather, the outofphase oscillator dynamics should be considered quantummechanically coherent, leading to the large entanglement entropies. In the following, we will study the direct implications of this entanglement.
Nuclear coordinate distribution shapes
We are now interested in the time evolution of the full nuclear coordinate distribution P_{n}(x_{n}, t) of molecule n and in particular, we will analyze the evolution of its tails in Fig. 4. In a product state ansatz [Eq. (7)], the instantaneous nuclear potential corresponds to a shifted harmonic oscillator. Then, nuclear wave packets of the individual molecules would always stay in a Gaussian shape. Crucially, this is not the case if we allow for finite entanglement. Then, in general, the Holstein coupling \(\propto \sum_n {\hat{x}}_{n}{\hat{\sigma }}_{n}^{+}{\hat{\sigma }}_{n}^{}\) does not factorize and thus modifies the wave packet shape over time (see e.g., Fig. 2e). To exemplify this, consider a single molecule n with a constant excitation fraction β (without cavity coupling). The time evolution under the Holstein Hamiltonian [Eq. (4)] leads to the following state at time t: \(\left{\phi }_{\beta }(t)\right\rangle =\sqrt{1\beta }{\left0\right\rangle }_{{{{{{{{\rm{exc}}}}}}}}}{\left0\right\rangle }_{{{{{{{{\rm{vib}}}}}}}}}+\sqrt{\beta }\exp [{{{{{{{\rm{i}}}}}}}}\phi (t)]{\left1\right\rangle }_{{{{{{{{\rm{exc}}}}}}}}}{\leftx(t)+{{{{{{{\rm{i}}}}}}}}p(t)\right\rangle }_{{{{{{{{\rm{vib}}}}}}}}}\) with the coherent state \({\left\alpha \right\rangle }_{{{{{{{{\rm{vib}}}}}}}}}=\exp (\alpha {\hat{b}}_{n}^{{{{\dagger}}} }{\alpha }^{* }{\hat{b}}_{n}){\left0\right\rangle }_{{{{{{{{\rm{vib}}}}}}}}}\) and the phase ϕ(t) due to the energy difference between states \({\left0\right\rangle }_{{{{{{{{\rm{exc}}}}}}}}}\) and \({\left1\right\rangle }_{{{{{{{{\rm{exc}}}}}}}}}\). For β ≠ 0, 1, this is generally an entangled state, and the shape of the nuclear wave packet (after tracing out the spin degree of freedom) is modified from the Gaussian shape, dependent on β.
In order to numerically study the shape of P_{n}(x_{n}, t) with our exact MPS method, we define its tails by \({x}_{n} \; < \; {x}_{{{{{{{{\rm{thr}}}}}}}}}^{{{{{{{{\rm{l}}}}}}}}}\) and \({x}_{n} \; > \; {x}_{{{{{{{{\rm{thr}}}}}}}}}^{{{{{{{{\rm{r}}}}}}}}}\), respectively. Here we choose a threshold value such that the tails of a ground state molecule include one percent of the weight \({\eta }_{0}=\{1\pm {{{{{{{\rm{erf}}}}}}}}[{x}_{{{{{{{{\rm{thr}}}}}}}}}^{{{{{{{{\rm{l}}}}}}}}/{{{{{{{\rm{r}}}}}}}}}]\}/2=1{0}^{2}\), which corresponds to \({x}_{{{{{{{{\rm{thr}}}}}}}}}^{{{{{{{{\rm{l}}}}}}}}/{{{{{{{\rm{r}}}}}}}}}\approx \mp 1.6\). We have confirmed that the underlying physics is generally independent of the specific choice of η_{0} for 0.1 > η_{0} > 10^{−4}, however, the relative magnitude of the changes generally increases for decreasing η_{0}. We define the timedependent tail weights:
In a simplified reaction picture, η^{l/r}(t) may be related to a reaction probability, e.g., for dissociation, if the coordinate x corresponds to the stretching of a critical bond in the system^{25}.
Without cavity, we can analytically solve the dynamics of the tail weights. For an initial single molecular excitation, the (N − 1) ground state molecules exhibit no dynamics, and the excited molecule oscillates between x_{1} = 0 and \({x}_{1}=2\sqrt{2}\lambda\) according to \({x}_{1}(t)=\sqrt{2}\lambda [1\cos (\nu t)]\). The tails are then given by \({\eta }^{{{{{{{{\rm{l}}}}}}}}/{{{{{{{\rm{r}}}}}}}}}(t)=(N1)\times {\eta }_{0}+\{1\pm {{{{{{{\rm{erf}}}}}}}}[{x}_{{{{{{{{\rm{thr}}}}}}}}}^{{{{{{{{\rm{l}}}}}}}}/{{{{{{{\rm{r}}}}}}}}}{x}_{1}(t)]\}/2\), with \({{{{{{{\rm{erf}}}}}}}}(x)=2\int\nolimits_{0}^{x}dz\exp ({z}^{2})/\sqrt{\pi }\) the error function. This is shown as gray dashed lines in Fig. 4.
The influence of cavity and disorder on the evolution of η^{l/r}(t) is shown in Fig. 4 for both an initial molecule excitation (Fig. 4a, b) and the cavity excitation scenario (Fig. 4c, d). We first discuss the disorderless case W = 0. For the molecule excitation (light dashdotted lines in Fig. 4a, b), we observe only a minimal modification from the nocavity case (gray dashed line). In contrast, for an initial cavity excitation (Fig. 4c, d), we find a strong suppression, in particular of the right tail weights due to the cavity. This is a manifestation of the polaron decoupling^{35}.
For W = g_{c}/2, in contrast, we find a distinctively different behavior. Focusing first on the right tail, we observe that disorder on average leads to a reduction of the tail at t ~ π/ν compared to the nocavity scenario, followed by an increase at later times, seen in Fig. 4b, d. This effect is significantly more pronounced for an initial cavity excitation (Fig. 4d) than for a molecule excitation (Fig. 4b). We attribute the dynamics observed in Fig. 4b, d to the outofphase oscillation of the different molecular vibrations (cf. Fig. 3). It implies that nuclear coordinates reach large values of x_{n} at different times and thus reduce the maximum weight of η^{r} at t ~ π/ν, but lead to a larger tail weight on average at later times. Importantly, we point again out that this outofphase oscillation should be considered as a quantum coherent process, i.e., the timedependent state is a large superposition where the vibrational degrees of freedom enter as linear superposition, as for the singlemolecule state \(\left{\phi }_{\beta }\right\rangle\), but with molecule and timedependent excitation fractions. The importance of vibrational entanglement for modeling the exact dynamics of the nuclear distribution is strikingly illustrated by the fact that product state simulations in Fig. 4 (dotted lines) fail to describe the correct dynamics.
We note that when we consider a time integration of the right tail weights, \({\eta }_{{{{{{{{\rm{avg}}}}}}}}}^{{{{{{{{\rm{r}}}}}}}}}=\int\nolimits_{0}^{2\pi /\nu }dt\,{\eta }^{{{{{{{{\rm{r}}}}}}}}}(t)\), i.e., the surface under the curves in Fig. 4b, d, we find that for large W, the exact \({\eta }_{{{{{{{{\rm{avg}}}}}}}}}^{{{{{{{{\rm{r}}}}}}}}}\) approximately agrees with the nocavity scenario. This phenomenon can be rationalized by the fact that, although the excitation is timedependently distributed over many molecules, in total there still only approximately remains one molecular excitation driving vibrational dynamics. Interestingly, this is not the case when time integrating the left tail weight, \({\eta }_{{{{{{{{\rm{avg}}}}}}}}}^{{{{{{{{\rm{l}}}}}}}}}\). In fact, this integrated weight increases compared to the nocavity case, which highlights the importance of the broadening and the nonGaussian shapes of the nuclear distributions.
Parameter scaling
Lastly, we want to systematically investigate the importance of the effects introduced in this paper as a function of disorder strength W, vibronic coupling strength ∝ λ, and molecule number N. In Fig. 5 we focus on the entanglement entropies and the right tail weights at time t = 2π/ν (red: initial molecule excitation, blue: initial cavity excitation). We find that the entanglement entropy S_{vib} and the right tail weight η^{r} scale extremely similarly with all parameters (comparing a to c and d to f in Fig. 5, respectively). This confirms the close relationship between both quantities. Furthermore, we find that for sufficiently large λ and W, the product state approximation (dotted lines in Fig. 5) breaks down completely and predicts only negligible modifications compared to the exact MPS simulations. This coincides with large values of S_{vib}, and thus underlines the essential role of entanglement between electrophotonic and vibrational degrees of freedom in the dynamics.
We observe that both S_{vib} and η^{r} grow with λ (see Fig. 5a, d). As discussed above, both entanglement and modifications to the right tail can be directly attributed to \({\hat{H}}_{{{{{{{{\rm{H}}}}}}}}}\), which scales with λ [Eq. (4)]. For small disorder W < g_{c}/2, i.e., in the strong coupling regime, we find that increasing disorder results in an increase of entanglement entropies and right tail weights (Fig. 5b, e), consistent with disorderenhanced excitation transfer. Interestingly, S_{vib} and η^{r} exhibit a peak between the weak and strong coupling limits. It becomes only weakly dependent on W in the weak coupling regime, i.e., for W > g_{c}. This behavior and the clear difference between excitation scenarios exemplify the rich physics in the intermediate coupling regime.
Strikingly, we also observe different scaling behaviors with the molecule number N between both initial states (Fig. 5c, f). For an individually excited molecule, the entanglement and the right tail weight decrease for large N [we subtract the ground state contribution (N − 1)η_{0} from the tail weight], in line with the analytical estimate for the scaling of excitation transfer between molecules ~Wt/N in the strong coupling regime. In contrast, for an initial cavityexcitation, the entanglement and tail weight remain approximately constant for large N. Here, the excitation transfer from the cavity to molecules occurs on the same timescale as Rabi oscillations, and the total amount of excitation transferred is perturbatively given by \({W}^{2}/{g}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) in the strong coupling regime, and thus to firstorder independent of N. This further highlights the important distinction between the two initial states, especially for large molecule numbers.
Conclusions
In summary, we have analyzed the coherent femtosecond dynamics in a disordered HTC model after incoherent photoexcitation. This minimal model features necessary ingredients for analyzing key quantum processes in polaritonic chemistry, including dynamics of electronic, vibrational, and photonic degrees of freedom^{35}. Using a matrix product state approach we have simulated the exact quantum manybody dynamics for realistic parameter regimes for mesoscopic system sizes. We have shown that disorderenhanced excitation transfer^{56,57,58}, both between the molecules and from the cavity to molecules, leads to coherent outofphase oscillations of the individual vibrational modes. Disorder thus strongly enhances the buildup of vibrational entanglement and modifications of the timedependent nuclear probability distributions, which are not captured in a product state (meanfield Ehrenfest) picture where electronic and nuclear degrees of freedom are treated as separable. We have highlighted that for large molecule numbers, an initial excitation in the cavity leads to much larger modifications than an initial molecular excitation. In general, disorderenhanced entanglement is a remarkable effect, since typically disorder is known to lead to a suppression of entanglement in various quantum manybody models^{63}.
Our results have direct implications for understanding the role of collective and quantummechanical effects in cavitymodified chemistry. While approximations based on wave functions that are separable between the electronic and the vibrational Hilbert space can provide useful insight into disorderfree systems, our work implies that the presence of disorder leads to a breakdown of such approaches. Our work emphasizes that for cavitymodified photochemistry with incoherent excitations, it is crucial to distinguish scenarios where the cavity or individual molecules are activated by the photon. The observation of largescale entanglement entropy buildup on very short femtosecond timescales suggests that quantum effects can play an important role in polaritonic chemistry experiments on timescales faster than the cavitydecay, and thus for experimentally feasible cavities with quality factors of Q ≳ 1000. Our work highlights the general importance of disorder for understanding polaritonic chemistry^{51,64,65}.
In the future, it will be interesting to consider more realistic molecular models, including beyondharmonic potential energy landscapes with more than one reaction coordinate, and featuring chemical reactions e.g., via electron transfer between multiple electronic levels, or conical intersections of energy surfaces. It will be interesting to extend our analysis to much longer times, when disorderenhanced transfer becomes even more relevant^{66}. It will be interesting to also investigate the effect of other sources of disorder in more detail, such as random molecular orientations and the cavity mode profile, leading to disordered cavity coupling constants (see Supplementary Note 1). Our numerical approach further allows us to also access regimes with multiple excitations, which will be an interesting regime to explore. Furthermore, our method can also easily include dissipative mechanisms, e.g., using a quantum trajectory approach,^{67} which has been proposed to lead to further modifications of the involved chemistry^{32,68,69,70,71}, an interesting prospect for future research.
Methods
Matrix product state method
We write the timedependent quantum state on the full electrophotonicvibrational Hilbert space in the form
Here, the different indices denote electronic excitation numbers for molecule n, i_{n} = 0, 1, the vibrational excitation number on molecule n, \({b}_{n}=0,\ldots ,{n}_{\max }^{{{{{{{{\rm{v}}}}}}}}}\) and the cavity mode occupation number, \(a=0,\ldots ,{n}_{\max }^{{{{{{{{\rm{p}}}}}}}}}\). While in principle \({n}_{\max }^{{{{{{{{\rm{v/p}}}}}}}}}\to \infty\), in practice the vibrational Hilbert space can be truncated at some reasonable occupation number. For this work we found that to capture all relevant physics of the tails of the nuclear coordinate distributions, \({n}_{\max }^{{{{{{{{\rm{v}}}}}}}}}=10\) is sufficient (see Supplementary Figure 2 and Note 5). Due to our choice of the initial state and the conservation of \({\sum }_{n}{\hat{\sigma }}_{n}^{+}{\hat{\sigma }}_{n}^{}+{\hat{a}}^{{{{\dagger}}} }\hat{a}\), furthermore we can set a photon cutoff at \({n}_{\max }^{{{{{{{{\rm{p}}}}}}}}}=1\) without any approximation. For N = 100 molecules, this implies a full Hilbert space size of 11^{N}2^{N+1} ≳ 10^{134}, clearly out of reach for any classical computer memory. In order to still make the highdimensional complex state tensor c amenable for storage in computer memory, we utilize a decomposition into products of smaller tensors, an MPS^{53}. In particular, we utilize an MPS with 2N + 1 tensors:
Here we introduced threedimensional tensors for the photonic, electronic, and vibrational degrees of freedom, \({{{\Gamma }}}_{a}^{{{{{{{{\rm{[p]}}}}}}}};{\alpha }_{0}{\alpha }_{1}}\), \({{{\Gamma }}}_{{i}_{n}}^{{{{[{{{{\rm{n}}}}_n}]}}};{\alpha }_{m}{\alpha }_{m+1}}\), and \({{{\Gamma }}}_{{b}_{n}}^{{{{[{{{{\rm{v}}}}_n}]}}};{\alpha }_{m}{\alpha }_{m+1}}\), respectively. The tensors are connected by the virtual indices α_{m} with m = 0, …, 2N + 1, and bond dimension χ (except for the edge indices, which are trivially α_{0} = α_{2N+1} = 1). The MPS can be brought, and updated, in a canonical form. Then, the virtual indices α_{m} correspond to an orthonormal basis, which is the eigenbasis of the reduced density matrix of the two blocks that the index connects^{53}. This effectively limits the entanglement entropy between the two blocks to \( < {\log }_{2}(\chi )\). For the MPS decomposition to become exact, one would need to choose very large values for \(\chi \sim \exp (N)\). However, limiting χ to computationally treatable magnitudes allows to effectively simulate dynamics on a truncated Hilbert space with restricted entanglement. In our simulations, we verified that all results converge with increasing χ, and therefore that our simulations capture all necessary entanglement and are quasiexact. In practice, we use χ = 128 for all plots (see Supplementary Figure 3 and Note 5). In our MPS form, tensors can be updated using the timeevolving Block decimation (TEBD) algorithm^{72}. Then, HTC coupling terms can be incorporated with nearestneighbor gate updates, while cavitycouplings can be incorporated using indexswap gates between the tensors and nearestneighbor gates. In practice, we choose a second order TEBD decomposition of the Hamiltonian with a time step of g_{c}/100, which we have verified to be sufficiently small for errors due to a finite time step to be negligible (see Supplementary Figure 4 Note 5). In the case of spinboson dynamics, TEBD in combination with swap gates has been previously shown to exhibit very wellbehaved convergence, which is preferable compared to updates that use variational concepts^{67}. Similarly, in order to compute S_{vib} we reorganize all vibrational degrees of freedom into a single block (using swap gates) and compute the entropy over the virtual index into that block. The excitation number conservation can be exploited to enhance the efficiency of tensor contractions and decompositions.
Data availability
The data of this study are available from the corresponding author, J.S., upon reasonable request.
Code availability
The code used for this study is available from the corresponding author, J.S., upon reasonable request.
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Acknowledgements
We are grateful to Felipe Herrera, Claudiu Genes, David Hagenmüller, Jerôme Dubail and Guido Masella for stimulating discussions. This work was supported by LabEx NIE (“Nanostructures in Interaction with their Environment”) under contract ANR11LABX0058 NIE and “ERANET QuantERA”  Projet “RouTe” (ANR18QUAN000501). This work of the Interdisciplinary Thematic Institute QMat, as part of the ITI 20212028 program of the University of Strasbourg, CNRS, and Inserm, was supported by IdEx Unistra (ANR10IDEX0002), SFRI STRAT’US project (ANR20SFRI0012), and EUR QMAT ANR17EURE0024 under the framework of the French Investments for the Future Program. We acknowledge support from ECOSCONICYT through project nr. C20E01 and the CNRS through the IEA 2020 campaign. G. P. acknowledges support from the Institut Universitaire de France (IUF) and the University of Strasbourg Institute of Advanced Studies (USIAS). Our MPS codes make use of the intelligent tensor library (ITensor)^{73}. Computations were carried out using resources of the HighPerformance Computing Center of the University of Strasbourg, funded by Equip@Meso (as part of the Investments for the Future Program) and CPER Alsacalcul/Big Data.
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D.W. and J.S. developed the conceptual approach and performed the simulations. D.W. derived the analytic estimates. J.S. developed the original code. G.P. and J.S. supervised the research. All authors analyzed and discussed the results and wrote the manuscript.
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Wellnitz, D., Pupillo, G. & Schachenmayer, J. Disorder enhanced vibrational entanglement and dynamics in polaritonic chemistry. Commun Phys 5, 120 (2022). https://doi.org/10.1038/s42005022008925
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DOI: https://doi.org/10.1038/s42005022008925
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